Hardy-Littlewood $k$-tuple Conjecture

By: Mahesh Dhulipala

Over the Summer and Fall of 2016, I worked with Dr. Hiary on modeling the error term of the Hardy-Littlewood $k$-tuple Conjecture. This conjecture estimates the number of prime constellations in a given range from $2$ to $N$ (where $N$ is a large positive integer). A prime constellation is defined as the set of prime numbers of the form $$(p,\, p+2m_1,\, p+2m_2,\,\, \ldots\,,\, p+2m_k)$$ where $m_1,m_2,\ldots,m_k$ are positive integers. The formula for the conjecture itself is given in the summary report. After some preliminary research, we guessed that the error term is bounded by something of the form $AN^{\alpha}$, where $A$ is a constant. To find the value of $\alpha$, we isolated it using the approximation: $$\alpha\approx \alpha_{\textrm{approx}} = \frac{\log|\textrm{Empirical} -\textrm{Conjecture}|}{\log N}.$$ The table 1 below shows the behavior of $\alpha_{\textrm{approx}}$ in the case of the $2$-tuple conjecture for various values of $N$ and $m_1$. Henceforth, we denote $m_1$ by $r$. As the graphs show, $\alpha$ seems to be bounded by $1/2$. (Note that the last three graphs are plotted on logarithmic scale.) Setting $\alpha=1/2$, we calculated the value of $A$ for various values of $N$. These results are presented in table 2 below, and they suggests that $A$ is inversely proportional to $N$, decaying with $N$ like an inverse power of $\log N$ perhaps. So $A$ is not a constant independent of $N$ after all, but bears some relation to it. This suggests that the initial guess for the upper bound, $AN^{\alpha}$, can be refined. Since we aimed to estimate the maximum error in the conjecture, we calculated the values of $A$ for various $N\ge 2^{20}$. The table of the resulting values of $A$ is given below in table 3.

Table 1: Graphs of $\alpha_{\textrm{approx}}$
$N$ $r$Graph
$2^{24}$1 to 100
$2^{28}$1 to 100
$2^{24}$ 2 to $2^{100}$
$2^{28}$2 to $2^{100}$
$2^{30}$ $2^{20}$ to $2^{30}$
Table 2: Behavior of $A$ ($r=2$)
$N$ $A$
$2^{20}$3.84914$\times 10^{-5}$
$2^{21}$1.98577$\times 10^{-5}$
$2^{22}$5.50649$\times 10^{-6}$
$2^{23}$8.85031$\times 10^{-6}$
$2^{24}$1.0991$\times 10^{-6}$
Table 3: Values of $A$ ($N=2^{20}$)
$r$ $A$
12.21274$\times 10^{-5}$
23.84914$\times 10^{-5}$
34.60216$\times 10^{-5}$
43.86429$\times 10^{-6}$
52.69231$\times 10^{-5}$
64.84824$\times 10^{-5}$
71.37947$\times 10^{-5}$
81.56882$\times 10^{-5}$
91.78449$\times 10^{-5}$
102.3626$\times 10^{-6}$
114.06328$\times 10^{-6}$
127.22066$\times 10^{-5}$
132.52207$\times 10^{-5}$
144.79284$\times 10^{-5}$
157.09623$\times 10^{-6}$
161.65303$\times 10^{-5}$
171.78241$\times 10^{-5}$
181.85252$\times 10^{-5}$
193.2703$\times 10^{-5}$
208.7101$\times 10^{-6}$
1003.54925$\times 10^{-5}$
10001.54595$\times 10^{-5}$